3. Given the parametric equations x = 3t /1 + t^3 and y = 3t^2 /1 + t^3 . a) Show that the curve produced also satisfies the equation x^3 + y^3 = 3xy. b) Compute the limits of x and y as t approaches −1: i. from the left ii. from the right c) To avoid the issue in part b), graph the curve twice in the same command, once for −5 < t < −1.5 and once for −0.5 < t < 5. d) Repeat part c) using the domains −100 < t < −1.5 and −0.5 < t < 100. In a print command, state how this plot is different from the plot in part c)

Prove that the proof by mathematical induction and the proof by strong induction are equivalent

Use Theorem 4.2.1 to determine which of the following sets of vectors are subspaces of F(−∞,∞). (Justify!)

(a) The set of all functions f ∈ F(−∞,∞) for which f(0) = 0.

(b) The set of all functions f ∈ F(−∞,∞) for which f(1) = 0.

(c) The set of all functions f ∈ F(−∞,∞) for which f(0) = 1.

(d) The set of all functions f ∈ F(−∞,∞) for which f(−x) = f(x) for all x ∈ R

Let M be the integer corresponding to the first letter of your last name. For example, if your last name begins with "A", M=1 and if your last name begins with "Z", M=26.

Let k=1/M and consider the logistic equation dy/dt = k y (M - y).

Construct a single figure including

1. Title "Logistic Equation Slope Field and Euler's Method solutions by FirstName LastName" with your actual first and last names, along the top
2. Labels for the axes
3. a slope field for -3 ≤ t ≤ 3, -M/2 ≤ y ≤ 3M/2
4. Euler's method solutions for initial conditions   y(0)=5M/4, y(0)=M, y(0)=M/2, y(0)=0 and y(0)=-M/4, in each case for -3 ≤ t ≤ 3, h=0.1
5. For each initial condition, a label "y(0) = (value)" but with the actual value, at an appropriate location
6. For each solution curve, a label "y(3) ≈ (Euler's method value)" but with the actual value, or "y(3) = ?", at an appropriate location

Last name starts with a P

1. Explain how to convert positive integers from base 3 to base 10 and vice versa. Include examples.

2. Explain how to convert fractions from base 3 to base 10 and vice versa. Include examples.

Prove that the equivalence classes of an equivalence relation form a partition of the domain of the relation. Namely, suppose ? be an equivalence relation on a set ? and define the equivalence class of an element ?∈? to be

[?]?:={?∈?|???}.

That is [?]?=?(?). Divide your proof into the following three peices:

1. Prove that every partition block is nonempty and every element ? is in some block.
2. Prove that if [?]?∩[?]?≠∅, then ???. Conclude that the sets [?]? for ?∈? are partitions of ?.
3. Prove that ???⟺[?]?=[?]?.

4. Below is a proof of Theorem 4.1.1 (b): For the zero vector ⃗0 in any vector space V and k ∈ R, k⃗0 = ⃗0.

   Justify for each of the eight steps why it is true.

   1. k⃗0+k⃗0=k(⃗0+⃗0)

   2. = k ⃗0

   3. k⃗0 is in V

   4. and therefore −(k⃗0) is in V .

   5. It follows that (k⃗0 + k⃗0) + (−k⃗0) = (k⃗0) + (−k⃗0)

   6. and thus k⃗0 + (k⃗0 + (−k⃗0)) = (k⃗0) + (−k⃗0).

   7. We conclude that k⃗0 + ⃗0 = ⃗0

   8. and so k⃗0 = ⃗0, as desired.

Indicate which of the following relations below are equivalence relations, strict partial orders, (weak) partial orders. If a relation is none of the above, indicate whether it is transitive, symmetric, or asymmetric. I won't be grading proofs of your results here, but I highly suggest you know how to prove your results.

1. The relation ?=?+1 between intgers ?,?.
2. The superset relation ⊇ on the power set of integers.
3. The divides relation on the nonnegative integers ?.
4. The divides relation on all the integers ?.
5. The divides relation on the positive powers of 4.

Use 3 steps of the Runge-Kutta (fourth order) method to solve the following differential equation to t = 2.4, given that y(0) = 2.3. In your working section, you must provide full working for the first step. To make calculations easier, round the tabulated value of y at each step to four decimal places.
a) Provide the four K-values that are calculated at the first step, with four decimal places. b) Provide your answer for y(2.4) with four decimal places.
(dy /dt) = 1.1ty

Use 4 steps of the Modified Euler’s method to solve the following differential equation to t = 2.6, given that y(0) = 1.1. In your working section, you must provide full working for the first two steps. To make calculations easier, round the calculations at each step to four decimal places, and provide your final answer with four decimal places. dy/ dt = 1.4sin(ty)

If the infimum of a bounded set E does not exist in E, then why must E have a limit point. And if the infimum does exist in E, why is it inconclusive?

The tank in the form of a right-circular cone of radius 7 feet and height 39 feet standing on its end, vertex down, is leaking through a circular hole of radius 4 inches. Assume the friction coefficient to be c=0.6 and g=32ft/s^2 . Then the equation governing the height h of the leaking water is

dhdt=_______________

If the tank is initially full, it will take _________ seconds to empty.

Let x1 > 1 and xn+1 := 2−1/xn for n ∈ N. Show that xn is bounded and monotone. Find the limit. Prove by induction